Find dy/dx if y = sin³(2x) cos(x²).

How to lay out the answer. Follow the frame, then write it in your own words.

1 · Write what is given and what is to be found. Given: y = sin³(2x) × cos(x²), a product. To find: dy/dx.
2 · Write the rules. Product rule (uv)′ = u′v + uv′, and the chain rule for each factor.
3 · Differentiate u = sin³(2x). u′ = 3 sin²(2x) × cos(2x) × 2 = 6 sin²(2x) cos(2x).
4 · Differentiate v = cos(x²). v′ = −sin(x²) × 2x = −2x sin(x²).
5 · Put them together. dy/dx = 6 sin²(2x) cos(2x) cos(x²) − 2x sin³(2x) sin(x²).
6 · Check and box the answer. Each term carries its inner derivative (2 and 2x). Box the result.

Scoring tip: Write u and v separately before combining. It stops the inner derivatives from getting lost.